Vector Calculations For Two-Plane Balancing 9-

If it were not for cross-effect, two-plane balancing could be accomplished in only three balancing runs or start-stop operations by making trial weight additions in both balancing planes at the same time, and constructing vector diagrams to get the proper solution.

Unfortunately, cross-effect is always present to some degree. Therefore, you can expect to use many balancing runs to get a good balancing using the single plane vector technique.

However, some machines may require from one-half hour to a full day for only one start-stop operation. On such machines, it would be most helpful to be able to minimize the number of balancing runs. When a considerable amount of time is required to start and stop a machine, or where severe cross-effect is encountered, the balancing problem can be greatly simplified by using the TWO-PLANE VECTOR METHOD.

In brief, the two-plane vector solution makes it possible to balance in two planes with only three start-stop operations. First, the original unbalance readings are recorded at the two bearings of the machine. Next, a trial weight is added to the first correction plane and the resultant readings at both bearings are again recorded. Finally, the trial weight is removed from the first correction plane and a trial weight added to the second correction plane. With this weight in the second plane, the resultant readings at both bearings are again noted and

recorded. Using the data recorded from the original and two trial runs, together with the known amount and location of the trial weights, a series of vector diagrams and calculations make it possible to eliminate the cross-effect of the system, and find both the amount and location of balance weight needed in each correction plane.

The two-plane vector solution requires from 15-30 minutes to complete. Therefore, it is essential that the data used be as accurate as possible. The most important readings taken are phase measurements. It is suggested that a phase reference card be used. This card can be made from a piece of vector paper and attached to cardboard of other stiff material. The center is cut out so that it may be held up to the center of shaft and the phase mark read directly. Other means such as a plastic phase reference card may be used. The calculation data sheet in Figure 28 has been developed for the two-plane vector calculation to serve as a guide and simplify recording of data. The Roman Numerals in the far left column correspond to the steps outlined in the detailed procedure below. The NEAR END (N) refers to the bearing observed; and the FAR END (F) refers to the opposite bearing and correction plane. Phase measurements for both the near end and far end must be taken using the same reference mark and phase reference card.

The procedure is as follows:

1. With the machine operating at the balancing speed and your analyzer filter properly tuned to a frequency that is equal to 1X RPM, observe and record the original phase for the near end (Item #1); the original amplitude for the near end (Item #2); the original phase for the far end (item #3), and the original amplitude for the far end

(Item #4).

2. Stop the machine and add a trial weight in the NEAR END correction plane. Record in (Item #5) the angular position of the trial weight in degrees clockwise from the

reference mark. (For example, with the trial weight in the position shown in Figure 29, we would record 240°.) Enter the amount of the trial weight as (Item # 6).

FIGURE 29. RECORD THE POSITION OF THE TRIAL WEIGHT IN DEGREES FROM THE REFERENCE MARK. HERE THE WEIGHT IS AT 240°

3. With the trial weight in the near end correction plane, operate the rotor at balancing speed. Observe and record the new phase for the near end

(Item #7); the new amplitude for the near end (Item #8); the new phase for the

far end (Item #9); and the new amplitude for the far end (Item #10).

4. Stop the machine and REMOVE the near end trial weight. Using the same weight or different weight if you prefer, add a trial weight at the far end

correction plane. Record as (Item #11) the position of the weight in degrees

clockwise from the reference mark (as viewed from the near end). Record the amount of this trial weight as (Item #12).

5. With the trial weight in the far end, again operate the rotor at the balancing speed. Observe and record the new phase reading for the near end (Item #13);

and new amplitude for the end (Item #14); the new phase reading for the far

end (Item #15); and the new amplitude for the far end (Item #16).

6. Using polar graph paper, construct vectors N, F, N2, F2, N3 and F3 by

drawing each at the observed phase angle, and to a length corresponding to the measured amplitude of vibration. For example, the vectors in Figure 30 have been drawn from the sample data in Figure 28.

NOTE: For accuracy, use the largest scale possible for constructing your vectors.

7. Construct vector A by drawing a line connecting the end of vector N to the end

of vector N2. See Figure 31. You will note on the data form in Figure 28 that

vector A is designated A = (N2). This notation is given to indicate the direction

of vector A and means that vector A is pointing from the end of vector N towards

the end of vector N2. This direction is very important for finding the angle of

vector A, (Item #17). The angle of vector A is found by transposing vector A

back to the origin of the polar graph as illustrated in Figure 31. A parallel ruler or set of triangles can be used to accurately transpose vector A parallel back

to the origin. For our example, the angle of vector A is 201° and is entered as

(Item #17). The amplitude of vector A, (Item #18) is found by simply measuring

its length using the same scale selected for vectors N, F, N2, etc. From our

example, Figure 31, vector A = 7.6 mils (193 microns).

Following the same procedure used to find the angle and amplitude of vector A,

proceed to find the values for vector B = (F3);

∝∝

Enter these values on the data form as (Items #19 through Items #24).

FIGURE 31

8. Do the calculations as indicated to find the values for (Items #25 through Items #32). Note that the numbers indicated in the “Calculation Procedures” column of

the data sheet are all referring to Item Numbers. Thus 25 = 21-17 means that the

value of Item #25 is found by subtracting the value of (Item #17) from the value of

(Item #21).

NOTE: During the calculations, you may find that some of your answers will be

negative (-) angles or angles larger than 360°. A negative angle, say -35°, may be converted to an equivalent positive angle by subtracting the angle from 360°. Thus 360°- 35°= 325°. An angle which is larger than 360° is converted to one less than 360° by subtracting 360° from the angle. For example, 463° - 360°= 103°.

9. Construct vectors

∝∝

vectors N, F, etc. The angle and length of vector

∝∝

calculated data, (Items #31 and #32) to construct vector BF.

10. Following the same procedure used to construct vectors A, B, etc., in step 7

above,proceed to construct vector C = (N BF) and vector D = (F

∝∝

enter the values for vectors C and D, (Items 33 through 36).

11. Calculate the values for (Items #37 and #38) following the same procedure

outlined for (Items #25 through #32) in step 8 above.

12. Using a new sheet of polar graph paper, construct the UNITY VECTOR (U), 1.0 unit

long at an angle of 0°. Note that the values for the unity vector have already been entered on the data form as (Items #39 and #40). The unity vector is always 1.0

unit at 0° for all two-plane vector problems. A suggested scale for the unity vector is 1.0 unit = 2.5 inches. (63.5 mm) See Figure 32.

NOTE: Do not confuse the UNITY VECTOR scale with that used to designate the

amplitude of vibration for vectors N, F, N2, etc. The unity vector can be thought of as a

dimensionless vector. This is why it is suggested that a separate sheet of graph paper be used, to help avoid confusion.

13. On the same graph paper with the unity vector, construct vector ∝∝B using the

same scale selected for the unity vector. The values for ∝∝B are obtained from

your calculated data, (Items #37 and #38). Remember, the value of vector ∝∝B

(Item #38) is expressed in units. Therefore, in the sample, Figure 32, ∝∝B = 0.22

units long at an angle of 311°.

14. Following the same procedure used to construct vectors A, B, etc, in step 7,

construct vector E =(∝∝B U).

Find and enter the values for vector E, (Items #41 and #42). Remember to

measure the length of ∝B, (Item #42) using the same unity scale.∝

15. Calculate the values for (Items 43 through #54), following the same procedure

outlined in step 8 above. (Items #51 and #52) Represent the position and

amount of the balance weight needed for the NEAR END correction plane. The angles for locating the balance weights are clockwise from the reference mark. 16. Before applying the new balance correction weights as indicated by (Items #51

through #54), it is suggested that a graphic check of your solution be made as

outlined below. This check will reveal whether or not any errors have been made in the solution.

A. On a new sheet of polar graph paper, construct vector O-A from your

calculated data (Items #43 and #44); and construct vector O/B from

(Items #45 and #46). For the length of vectors, use the same scale

selected for your original vectors N, F, N2, etc.

B. Calculate the amplitude and angle values for vector ΟΟ/BB. Amplitude

= (Item #50 x Item #24); and the angle = (Item #49 + Item #21).

C. Calculate the amplitude and angle values for vector ΟΟ-

∝∝

= (Item #40 x Item #22); and the angle = (Item (#47 + Item #21).

D. Using the calculated values, proceed to construct vectors ∅∅BB and ∅

∅A to the same scale used for ∅∅B and ∅∅A. See Figure 33.

E. Construct vector X by adding vector ∅∅A and ∅∅BB. This is done by

completing the parallelogram as shown in Figure 34. The diagonal of this parallelogram is vector X that should be equal in length but directly opposite the original N vector.

F. Construct vector Y by adding vectors ∅∅/B and ∅∅-

∝∝

completing the parallelogram. Vector Y should be equal in length and directly opposite your original F vector. See Figure 34.

G. If vectors N and X or vectors F and Y are not equal and opposite, this

17. If the graphic check indicates that your solution has been done correctly, proceed to make the balance corrections as indicated in step 15. Be sure the trial weight added in step 9 has been removed.

18. With the balance corrections applied, operate the rotor and check to be sure the vibration has been reduced to an acceptable level.

FIGURE 33. GRAPHIC CHECK SOLUTION

19. If the applied corrections significantly reduced the unbalance, yet further

correction is required, observe and record the new unbalance data - amplitude and phase - for the near and far ends. Enter this data as items 1 through 4 on a new two-plane vector data sheet. Also enter on the new form those items marked by an asterisk (*) from the original data (i.e. items 5, 6, 11, 12, 17, 18, 19, 21, 25, 26, 27, 28, 41 and 42). Now, simply recalculate items 29 through 36 and 43 through 54 to find the additional balance corrections required. Do not disturb your previous corrections. Here also, the GRAPHIC CHECK can be performed to verify that your solution is correct before applying the additional corrections. The procedure for applying further balance corrections can be of great value if this rotor should require rebalancing any time in the future.

Simply attach the vibration pickups in the same position used during the original balancing, and take phase readings using the same reference mark. Enter the new unbalance data on the data sheet as items 1 through 4. From the original balance data, enter those items marked by an asterisk (*) and simply recalculate to find the new required balance corrections. In summary, once the two-plane vector calculation has been worked successfully for a particular rotor, this rotor can be balanced in two-planes in the future in only one run.